Hermitian Operators are Self-Adjoint

By definition, if $A$ is an Hermitian Operator then $A^{\ast }=A$. In particular, $A^{\ast }$ had the same domain as $A$, so $A:⟨Ax,y⟩=⟨x,Ay⟩,\forall x,y\in H$, making $A$ a Self-Adjoint Operator operator. The reverse is not true: Not all self-adjoint operators are hermitian.